Question

The report referenced in the previous exercise also stated that the proportion who thought their parents would help with buying a house or renting an apartment for the sample of young adults was 0.37 . For the sample of parents, the proportion who said they would help with buying a house or renting an apartment was 0.27 . Based on these data, can you conclude that the proportion of parents who say they would help with buying a house or renting an apartment is significantly less than the proportion of young adults who think that their parents would help?

Short answer

Depending on the calculated p-value, we either reject or fail to reject the null hypothesis. If the p-value is less than 0.05, then we reject the null hypothesis and conclude that there is a significant difference between the proportion of young adults who think their parents would help versus the proportion of parents who say they would help. If the p-value is greater than 0.05, then we fail to reject the null hypothesis and conclude that there is not a significant difference.

Step-by-step solution

Calculating the Pooled Proportion

The pooled proportion is the total 'successes' divided by the total sample size. Here, a 'success' means help was provided. Suppose that each sample has a size of n. Then, the pooled proportion, p, can be calculated like this: p = [p1*n + p2*n] / [2*n]. Using the given proportions 0.37 (p1) and 0.27 (p2), we get p = (0.37n + 0.27n) / (2n) = 0.32.

Computing the Test Statistic (Z Score)

The test statistic for a hypothesis test for two proportions is a Z-score. The formula for the test statistic is Z = (p1 - p2) / sqrt [p(1-p){(1/n1)+(1/n2)}]. Using 0.37 as p1, 0.27 as p2, 0.32 for p, and assuming sample sizes n1 = n2 = n, we substitute these values into the formula to get Z.

Finding the P-Value

Once we have the Z score, we determine the p-value. The p-value is the probability that a variable would be observed as extreme as the test statistic assuming the null hypothesis is true. If the p-value is smaller than the significance level (usually 0.05), then we reject the null hypothesis. In our case, we look up the P(Z > obtained z score) in a standard normal (Z) distribution table.

Making the Decision

After obtaining the p-value, we compare it to our significance level (commonly 0.05). If the p-value is smaller than 0.05, we reject the null hypothesis and conclude that there is a significant difference between two populations.